MinT - Best Known (30, 30+4, s)-Nets in Base 5

Best Known (30, 30+4, s)-Nets in Base 5

(30, 30+4, large)-Net over F₅ — Constructive and digital

Digital (30, 34, large)-net over F₅, using

net defined by OOA [i] based on linear OOA(5³⁴, large, F₅, 4, 4), using
- appending kth column [i] based on linear OOA(5³⁴, large, F₅, 3, 4), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(5², 6, F₅, 3, 2) (dual of [(6, 3), 16, 3]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(3;16,5) [i]
    2. linear OOA(5³², 8388601, F₅, 3, 4) (dual of [(8388601, 3), 25165771, 5]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(5³², 8388602, F₅, 2, 4) (dual of [(8388602, 2), 16777172, 5]-NRT-code), using
        trace code [i] based on linear OOA(25¹⁶, 4194301, F₂₅, 2, 4) (dual of [(4194301, 2), 8388586, 5]-NRT-code), using
        OOA 2-folding [i] based on linear OA(25¹⁶, 8388602, F₂₅, 4) (dual of [8388602, 8388586, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(25¹⁶, large, F₂₅, 4) (dual of [large, large−16, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(30, 30+4, large)-Net in Base 5 — Upper bound on s

There is no (30, 34, large)-net in base 5, because

2 times m-reduction [i] would yield (30, 32, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]